Isomorphic iff potentially conjugate

Statement

For just one pair of isomorphic subgroups

Suppose $G$ is a group and $H, K \leq G$ are isomorphic subgroups, i.e., there is an isomorphism of groups from $H$ to $K$ (Note that this isomorphism need not arise from an automorphism of $G$ , so $H$ and $K$ need not be automorphic subgroups). The,n there exists a group $L$ containing $G$ such that $H, K$ are conjugate subgroups inside $L$ .

For a collection of many pairs of isomorphism subgroups

Suppose $G$ is a group, $I$ is an indexing set, and $H_{i} ≅ K_{i}$ are pairs of isomorphic subgroups of $G$ for each $i \in I$ . Then, there exists a group $L$ containing $G$ as a subgroup such that $H_{i}$ and $K_{i}$ are conjugate subgroups in $L$ for each $i \in I$ . (Note: The choice of conjugating element may differ for different $i \in I$ ).

Moreover, there is a natural construction of such a group $L$ , called a HNN-extension. In the case that $G$ is a torsion-free group, we can ensure

Related facts

Facts about automorphisms extending to inner automorphisms

Inner automorphism to automorphism is right tight for normality: In other words, if $σ$ is an automorphism of $G$ , there exists a group $L$ containing $G$ as a normal subgroup, and an inner automorphism of $L$ whose restriction to $G$ equals $σ$ .
Left transiter of normal is characteristic: A direct application of the fact that any automorphism of a group extends to an inner automorphism in a bigger group containing it as a normal subgroup. This says that $H \leq G$ is such that (whenever $G$ is normal in $L$ , $H$ is also normal in $L$ ) if and only if $H$ is characteristic in $G$ .
Characteristic of normal implies normal

Applications

Elements of same order are potentially conjugate: If $x, y \in G$ are such that $x, y$ have the same order, then there is a group $L$ containing $G$ in which $x$ and $y$ are conjugate elements.
Every group is a subgroup of a group with two conjugacy classes
Every group is a subgroup of a simple group
Every torsion-free group is a subgroup of a torsion-free group with two conjugacy classes